Theorem cbviunv

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003) Add disjoint variable condition to avoid ax-13 . See cbviunvg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypothesis	cbviunv.1	|- ( x = y -> B = C )
Assertion	cbviunv	|- U_ x e. A B = U_ y e. A C

Proof

Step	Hyp	Ref	Expression
1		cbviunv.1	|- ( x = y -> B = C )
2		nfcv	|- F/_ y B
3		nfcv	|- F/_ x C
4	2 3 1	cbviun	|- U_ x e. A B = U_ y e. A C